# Is this statement about plots in statistics true?

I'm doing a project. But I've gotten so many different answers and numbers on the internet. Minitab, Excel, and a few of those smart online calculators all gave me different answers on the quartiles by a few decimal places with the exact same data. Mostly in the tens decimal place.

But the main thing is if my paragraph explaining my graph is true? I'm mainly confused on the outliers and whiskers. My math should be right (again, different answers), but if those are indeed my whiskers, I don't have any outliers since my data is within that range?

A boxplot gives you information about the shape, median, and quartiles of your distribution. In figure 2, the shape is just slightly skewed left, the first quartile is 66.88, the median is 68.2, and the third quartile is 69.55, the lower whisker is 63.4, and the upper whisker is 71.9. To calculate any outliers you must first find the inner quartile range or IQR, which is Quartile 3- Quartile 1. My IQR is 2.67. To find lower outliers you use Quartile 1-(1.5*IQR). Which in my case is 62.88. To calculate any upper outliers you use Quartile 3+(1.5*IQR). Which in my case is 73.56. My boxplot doesn’t have any outliers since my whiskers are within this range.

I'm so confused with everything online. ## 1 Answer

About ten (slightly) different definitions of quartile are in common usage in various computer programs and textbooks. In particular, there are different ways to treat gaps and ties. For small $n$ these differences can result in noticeably different boxplots, and I don't think boxplots should be used for sample sizes less than about 15 or 20. (The idea of using quartiles is to partition sorted data into four approximately equal 'chunks'. But if $n = 9$ or $13,$ compromises must be made.)

In your case, it seems you're using larger samples. So the differences among methods are not likely to give boxplots that differ in important ways. (Even so, some people have enjoyed contriving examples where different quantile rules make the difference whether a particular observation is flagged as an 'outlier'.)

If you are mainly focused on whether your dataset has outliers or not, you should realize that data from normal and other commonly encountered distributions naturally produce outliers. An 'outlier' is an observation that is worth a second look: Is it a data entry error? Result of equipment malfunction? And so on. But the answer may very well be that there is nothing 'wrong' with the outlier. It is a serious mistake systematically to delete or ignore outliers unless they are quite extreme or they arise from an identifiable error in measurement or transcription.

Note: If you want to explore technical differences among quantile rules a good place to start is with the page in R statistical software that is summoned by typing ? quantile in the Session Window. (Nine different commonly used rules are detailed, along with notes about which software uses which one. Another issue is Tukey's 'fourths' often used in making boxplots; they are similar to quartiles, but are not really quartiles.)

• But is what I said true? Like my paragraph? Like consistent with my data? I guess I don't truly understand the difference between an outlier and a whisker. Because a whisker can be an outlier. But in this case since the outlier calculations are bigger and lower than my maximum and minimum values, does that mean I don't have outliers? And are my maximum and minimum in this case, my lower and upper whiskers? Thank you for your help! – MorbidTag Apr 18 '17 at 20:34
• Yes. If you had errors there I would have noted them. By definition, whiskers stop short of outliers. If no outliers, whiskers reach down to min and up to max. If there are outliers (beyond fences at Q1-1.5IQR and Q3+1.5IQR), then whiskers extend up or down to most extreme values inside fences. – BruceET Apr 18 '17 at 21:26